Hyperbolic graphs of small complexity
Abstract.
In this paper we enumerate and classify the “simplest” pairs where is a closed orientable manifold and is a trivalent graph embedded in .
To enumerate the pairs we use a variation of Matveev’s definition of complexity for manifolds, and we consider only irreducible pairs, namely pairs such that any 2sphere in intersecting transversely in at most points bounds a ball in either disjoint from or intersecting in an unknotted arc. To classify the pairs our main tools are geometric invariants defined using hyperbolic geometry. In most cases, the graph complement admits a unique hyperbolic structure with parabolic meridians; this structure was computed and studied using Heard’s program Orb and Goodman’s program Snap.
We determine all irreducible pairs up to complexity 5, allowing disconnected graphs but forbidding components without vertices in complexity 5. The result is a list of 129 pairs, of which 123 are hyperbolic with parabolic meridians. For these pairs we give detailed information on hyperbolic invariants including volumes, symmetry groups and arithmetic invariants. Pictures of all hyperbolic graphs up to complexity 4 are provided. We also include a partial analysis of knots and links.
The theoretical framework underlying the paper is twofold, being based on Matveev’s theory of spines and on Thurston’s idea (later developed by several authors) of constructing hyperbolic structures via triangulations. Many of our results were obtained (or suggested) by computer investigations.
2000 Mathematics Subject Classification:
Primary 57M50; Secondary 57M27, 05C30, 57M20.1. Introduction
The study of knotted graphs in manifolds is a natural generalization of classical knot theory, with potential applications to chemistry and biology (see e.g. [10]). In knot theory, extensive knot tables have been built up through the work of many mathematicians (see e.g. Conway [6] and Hoste–Thistlethwaite–Weeks [21]). There has been much less work on the tabulation of knotted graphs, but some knotted graphs in have been enumerated in order of crossing number by Simon [45], Litherland [27], Moriuchi [38, 39], and Chiodo et. al. [5].
In this paper we classify the simplest trivalent graphs in general closed manifolds. We first enumerate them using a notion of complexity which extends Matveev’s definition for manifolds [32], and then we classify them with the help of geometric invariants, mostly defined using hyperbolic geometry.
More precisely, the objects considered in this paper are pairs where is a closed, connected orientable manifold and is a trivalent graph in . The graph may contain loops and multiple edges, and is possibly disconnected (in particular, can be a knot or a link). To avoid “wild” embeddings we work in the piecewise linear category: thus is a PLmanifold and is a 1dimensional subcomplex, and we aim to classify graphs up to PLhomeomorphisms of pairs.
Following [32], a compact polyhedron is called simple if the link of every point of embeds in the 1skeleton of the tetrahedron (the complete graph with 4 vertices). Points having the whole of this graph as a link are called vertices of . Moreover, as defined in [41], is a spine of a pair if it embeds in so that its complement is a finite union of balls intersecting in the simplest possible ways, as shown in Figure 1.
As usual in complexity theory, the complexity is then defined as the minimal number of vertices in a simple spine of . The case considered in [41] is actually that of 3orbifolds, but the definition of complexity is the same as just given, except that a contribution of the edge labels is also introduced. When we recover the original definition of Matveev, thus getting the equality . In general, we have .
For manifolds, Matveev showed that complexity is additive under connected sum and that it behaves particularly well on irreducible manifolds (i.e. manifolds in which every 2sphere bounds a 3ball). In particular, there exist only finitely many irreducible manifolds with given complexity. These facts extend to the context of the pairs described above, with the following notion of irreducibility: is irreducible if every 2sphere embedded in and meeting transversely in at most two points bounds a ball intersecting as in Figure 1, left or centre (in particular, there exists no 2sphere meeting in one point).
This paper is devoted to the enumeration and the geometric investigation of all irreducible graphs of small complexity. As usual in dimensional topology, a key role in the study of our graphs is played by invariants coming from hyperbolic geometry, which in particular provided the tools we used in most cases to distinguish the pairs from each other.
While the complement of in very often has no hyperbolic structure with geodesic boundary (for instance, it is often a handlebody), most pairs are indeed hyperbolic in a more general sense, namely they are hyperbolic with parabolic meridians. This means that carries a metric of constant sectional curvature which completes to a manifold with noncompact geodesic boundary having:

toric cusps at the knot components of ,

annular cusps at the meridians of the edges of , and

geodesic 3punctured boundary spheres at the vertices of .
This hyperbolic structure is the natural analogue of the complete hyperbolic structure on a knot or link complement and is also useful when studying orbifold structures on .
By MostowPrasad rigidity, a hyperbolic structure with parabolic meridians is unique if it exists, so its geometric invariants only depend on . One can therefore use the volume and Kojima’s canonical decomposition [24, 25] to distinguish hyperbolic graphs. For the pairs in our list we have constructed and analyzed the hyperbolic structure using the computer program Orb, written by the first named author [19].
Since knots and links have already been widely studied in many contexts, this paper focuses mostly on graphs containing vertices.
Number of hyperbolic graphs
Table 1 gives a summary of our results. Up to complexity our census of hyperbolic graphs is complete and contains elements, consisting of 5 knots, 24 graphs, 13 handcuffs, and 3 distinct connected graphs with four vertices. The graph types occurring are shown in Figure 2. In complexity 5 we decided to rule out knot components, and we found 78 more hyperbolic graphs. Out of our 123 graphs, 36 lie in .
type  

knot (in )  0 (0)  0 (0)  1 (1)  4 (1)  – (–) 
(in )  0 (0)  2 (1)  4 (1)  18 (4)  49 (10) 
(in )  1 (1)  1 (0)  3 (2)  8 (2)  27 (8) 
(in )  0 (0)  1 (1)  0 (0)  0 (0)  2 (2) 
(in )  0 (0)  0 (0)  0 (0)  1 (1)  0 (0) 
(in )  0 (0)  0 (0)  0 (0)  1 (1)  0 (0) 
Complexity and volume
As shown in Table 1, there is a single hyperbolic graph of smallest complexity . It is a handcuff graph in , described in Figure 4 and Example 2.1. It is also the hyperbolic graph with vertices of least volume 3.663862377… This fact confirms the following relationships between complexity and hyperbolic geometry, which have already been verified for closed manifolds [32, 15], cusped manifolds [2, 3, 14, 15], and manifolds with arbitrary (geodesic) boundary [13, 26, 36, 11]:

Objects having complexity zero are not hyperbolic.

Among hyperbolic ones, the objects having lowest volume have the lowest complexity.
Note that complexity and volume may share the same first segments of hyperbolic objects (as they do) but are qualitatively different globally, because in general there are finitely many hyperbolic objects of bounded complexity, while infinitely many ones may have bounded volume thanks to Dehn surgery.
Compact totally geodesic boundary
It may happen that has a hyperbolic metric which completes to a manifold with compact totally geodesic boundary. In this case we say that is hyperbolic with geodesic boundary, which implies that is also hyperbolic (with parabolic meridians), but as mentioned above the converse is often false. By analyzing the graphs in Table 1, we have established the following:
Nonhyperbolic graphs
The (0,1,2)irreducible graphs of complexity 0 were detected by theoretical means, see Section 3. There are 3 knots (cores of Heegaard tori in , and ) and the trivial graph in , and they are all nonhyperbolic. In complexity we have classified all (0,1,2)irreducible nonhyperbolic graphs, finding only 16 knots and two links. The same phenomenon happens for , where we have shown that only knots and links are (0,1,2)irreducible and nonhyperbolic. However we refrained from classifying them completely, confining ourselves to those in with . Since our primary interest was in hyperbolic graphs, we decided to rule out knot components in complexity , but quite interestingly we have found some nonhyperbolic examples in this case. Our results are summarized by Table 2 and the next statement:
Proposition 1.2.
The only irreducible nonhyperbolic graphs with such that has no knot component are the trivial graph in , which has complexity , and five pairs in complexity , where is a graph and contains an embedded Klein bottle.
type  

knot (in )  3 (1)  4 (1)  12 (1)  – (4)  – (–)  – (–) 
2link (in )  0 (0)  1 (1)  1 (0)  – (1)  – (–)  – (–) 
2t (in )  1 (1)  0 (0)  0 (0)  0 (0)  0 (0)  5 (0) 
Some open problems
We conclude this introduction by suggesting a few problems for further investigation.

Enumerate the first few hyperbolic graphs with parabolic meridians in order of increasing hyperbolic volume.

Enumerate the first few hyperbolic manifolds of finite volume with (compact or noncompact) geodesic boundary in order of increasing hyperbolic volume.

Enumerate the first few closed hyperbolic 3orbifolds in order of increasing complexity as defined in [41].

Enumerate the first few closed hyperbolic 3orbifolds in order of increasing hyperbolic volume.

Determine the exact complexity of infinite families of knotted graphs, for example the torus knots in lens spaces (see Conjecture 6.4 below).
Note that Kojima and Miyamoto [26, 36] have already identified the lowest volume hyperbolic manifolds with compact and noncompact geodesic boundary. Perhaps the “Mom technology” introduced by Gabai, Meyerhoff and Milley [14, 15] may offer an approach to (1) and (2). Recent work of Martin with Gehring and Marshall [16, 29] has identified the lowest volume orientable hyperbolic 3orbifold.
2. Hyperbolic geometry
In this section we review the main geometric notions and results we will need in the rest of the paper.
2.1. Hyperbolic structures with parabolic meridians
To help classify knotted graphs, we will study hyperbolic structures analogous to the compete hyperbolic structure on the complement of a knot or link. Given a graph in a closed orientable manifold , let be the manifold obtained from by removing an open regular neighbourhood of the vertex set of . Thus is a noncompact manifold with boundary consisting of 3punctured spheres, one corresponding to each vertex of . Then we say that has a hyperbolic structure with parabolic meridians if admits a complete hyperbolic metric of finite volume with geodesic boundary (with toric and annular cusps). Equivalently, the double of admits a complete hyperbolic metric of finite volume (with toric cusps). Such a hyperbolic structure on is unique by a standard argument using MostowPrasad rigidity [46] and Tollefson’s classification [49] of involutions with 2dimensional fixed point set (see [47] and also [12]).
Example 2.1.
The simplest hyperbolic handcuff graph can be obtained from one tetrahedron with the two front faces folded together and the two back faces folded together giving a triangulation of with the graph contained in the 1skeleton as shown in Figure 4.
If we truncate the vertices of the tetrahedron until all edge lengths are zero, the result can be realized geometrically by a regular ideal octahedron in hyperbolic space, as shown in Figure 5. We can then glue the 4 unshaded faces together in pairs so that the other 4 shaded faces form two totally geodesic 3punctured spheres.
This gives a hyperbolic structure with parabolic meridians for with hyperbolic volume 3.663862377… The work of Miyamoto and Kojima [26, 36] shows that this is the smallest volume for trivalent graphs. Their work also implies that a trivalent graph having this volume is obtained by identifying the unshaded faces of an ideal octahedron as above, and hence has complexity . Therefore the handcuff graph in Figure 4 is the unique graph of minimal volume.
We next describe topological conditions for the existence of a hyperbolic structure with parabolic meridians. Let denote the graph exterior, i.e., the compact manifold obtained from by removing an open regular neighbourhood of the graph . Then is a disjoint union of pairs of pants (corresponding to the vertices of ) and a collection of annuli and tori (corresponding to the edges and knots in ). Thurston’s hyperbolization theorem for pared manifolds [37, 23] implies the following:
Theorem 2.2.
admits a hyperbolic structure with parabolic meridians if and only if

is irreducible and homotopically atoroidal,

consists of incompressible annuli and tori,

there is no essential annulus , and

is not a product where is a pair of pants.
This hyperbolic structure is unique up to isometry.
Remark 2.3.
To obtain a hyperbolic structure with geodesic boundary on a general pared manifold , we would need to add the requirements that incompressible and is acylindrical (i.e., every annulus is homotopic into ). But these conditions follow here since consists of punctured spheres (see [1, pp. 243244]).
The conditions for hyperbolicity simplify considerably when is irreducible, as defined in the introduction. To elucidate the notion, we say that is:

0irreducible if every 2sphere in disjoint from bounds a 3ball in disjoint from ;

1irreducible if there exists no 2sphere in meeting transversely in a single point;

2irreducible if every 2sphere in meeting transversely in two points bounds a ball in that intersects in a single unknotted arc.
Then a graph is irreducible if it is irreducible for .
Theorem 2.4.
admits a hyperbolic structure with parabolic meridians if and only if

is irreducible,

is homotopically atoroidal and is not a solid torus or the product of a torus with an interval, and

is not the trivial graph in .
Proof.
It is easy to check that the conditions listed are necessary for hyperbolicity. To show that they are sufficient, first note that irreducibility of implies that is irreducible, and irreducibility implies that is incompressible or is a solid torus, but the latter possibility is excluded. Moreover is not a product where is a pair of pants, because is not the trivial graph in . According to the previous theorem we are only left to show that there cannot exist an essential annulus . Suppose the contrary and note that each of the two components of is incident to either an annular or a toric component of . We show that the existence of such an annulus is impossible by considering the three possibilities:

If is only incident to annuli of , we readily see that irreduciblity is violated.

If is incident to an annular component of and a torus component of , then the boundary of a regular neighbourhood of is another annulus incident to only. Again we see that irreducibility is violated, since the resulting sphere does not bound a ball containing a single unknotted arc.

If is incident to toric components only, proceeding as in the previous case we find one or two tori, depending on whether the toric components are distinct or not. Homotopic atoroidality implies that these tori must be compressible or boundary parallel in . Using irreducibility of and incompressibility of , we find that is Seifert fibred with the core circle of as a fibre and base space either a pair of pants, an annulus with at most one singular point, or a disc with at most two singular points. By homotopic atoroidality, we deduce that is the product of a torus and an interval or a solid torus, contrary to our assumptions.
∎
Corollary 2.5.
If is a trivalent graph containing at least one vertex, then is hyperbolic with parabolic meridians if and only if is irreducible, geometrically atoroidal and not the trivial graph in .
2.2. Hyperbolic structures with geodesic boundary
Let be a graph, and let denote the graph exterior as above. Let us define as the manifold obtained by mirroring in its nontoric boundary components, so is either closed or bounded by tori. Then minus its toric boundary components has a hyperbolic structure with totally geodesic boundary if and only if the interior of has a complete hyperbolic structure. By Thurston’s hyperbolization theorem [37, 23] and MostowPrasad rigidity (see [47, p. 14]) we then have:
Theorem 2.6.
minus its toric boundary components admits a hyperbolic structure with totally geodesic boundary if and only if is irreducible, boundary incompressible, homotopically atoroidal, and acylindrical. This hyperbolic structure is unique up to isometry.
2.3. Hyperbolic orbifolds
One of the initial motivations of our work was the study of hyperbolic orbifolds, but the analysis of graphs turned out to be interesting enough by itself, so we decided to leave orbifolds for the future. However we mention them briefly here.
Given a trivalent graph in a closed manifold , we obtain an orbifold associated to by attaching an integer label to each edge or circle of . Note that we do not impose any restrictions on the labels of the edges incident to a vertex , so from a topological viewpoint gives rise either to an interior point of (if ) or to a boundary component of — a 2orbifold of type .
We will say that is hyperbolic if admits an incomplete hyperbolic metric whose completion has a cone angle along each edge or circle in . Depending on whether is positive, zero or negative, a vertex with incoming labels gives rise to an interior point of to which the singular metric extends, to a cusp of , or to a totally geodesic boundary component of .
The main connections between orbifold hyperbolic structures and those we deal with in this paper are as follows:

If has a hyperbolic orbifold structure for some choice of labels , then admits a hyperbolic structure with parabolic meridians.

If admits a hyperbolic structure with parabolic meridians then the corresponding orbifolds are hyperbolic provided all labels are sufficiently large; moreover the structure with parabolic meridians can be regarded as the limit of the orbifold hyperbolic structures as all labels tend to infinity.
2.4. Algorithmic search for hyperbolic structures
As already mentioned, the hyperbolic structures and related invariants on the 123 pairs of our census have been obtained using the computer program Orb [19]. More details on this program will be provided below, but we outline here the underlying theoretical idea (due to Thurston [46]) of the algorithmic construction of a hyperbolic structure with geodesic boundary on a pared manifold , where is compact but not closed and is a collection of tori and annuli on .
The starting point is a (suitably defined) ideal triangulation of , namely a realization of as a gluing of generalized ideal tetrahedra. Each of these is a tetrahedron with its vertices removed and, depending on its position with respect to and , perhaps entire edges and/or open regular neighbourhoods of vertices also removed. The next step is to choose a realization of each of these tetrahedra as a geodesic generalized ideal tetrahedron in hyperbolic space. These realizations are parameterized by certain moduli, and the condition that the hyperbolic structures on the individual tetrahedra match up to give a hyperbolic structure on translates into equations in the moduli. The algorithm then consists of changing the initial moduli using Newton’s method until the (unique) solution of the equations is found.
When is closed one can search for its hyperbolic structure using a similar method, starting from a decomposition of into compact tetrahedra [4].
2.5. Canonical cell decompositions
Whenever a hyperbolic manifold is not closed, it admits a canonical decomposition into geodesic hyperbolic polyhedra, which allows one to very efficiently compute its symmetry group and compare it for equality with another such manifold. The decomposition was defined by Epstein and Penner [9] when but has cusps, and by Kojima [24, 25] when . We will now briefly outline the latter construction.
Begin with the geodesic boundary components of and very small horospherical cross sections of any torus cusps of , and expand these surfaces at the same rate until they bump to give a 2complex (the cut locus of the initial boundary surfaces). Then dual to this complex is the Kojima canonical decomposition of into generalized ideal hyperbolic polyhedra. This is independent of the choice of horosphere cross sections provided they are chosen sufficiently small, and gives a complete topological invariant of the manifold.
Thus two finite volume hyperbolic manifolds with geodesic boundary are isometric (or, equivalently, homeomorphic) if and only if their Kojima canonical decompositions are combinatorially the same; and the symmetry group of isometries of such a manifold is the group of combinatorial automorphisms of the canonical decomposition. Similarly, two graphs admitting hyperbolic structures with parabolic meridians are equivalent if and only if there is a combinatorial isomorphism between their canonical decompositions taking meridians to meridians; and the group of symmetries of such a graph is the group of combinatorial automorphisms of the canonical decomposition taking meridians to meridians.
2.6. Arithmetic invariants
Let us first note that a hyperbolic structure on an orientable manifold without boundary corresponds to a realization of the manifold as the quotient of hyperbolic space under the action of a discrete group of orientationpreserving isometries of . If the manifold has boundary, should be replaced by a invariant intersection of closed halfspaces in . Moreover for any given hyperbolic manifold, the group is welldefined up to conjugation within the full group of orientationpreserving isometries of , which is isomorphic to .
If is a discrete subgroup of , then the invariant trace field is the field generated by the traces of the elements of lifted to . This is a commensurability invariant of (unchanged if is replaced by a finite index subgroup). Further, if has finite volume then it follows from MostowPrasad rigidity that is a number field, i.e., a finite degree extension of the rational numbers . (See [28] for an excellent discussion and proofs.)
If a trivalent graph admits a hyperbolic structure with parabolic meridians, then is the convex hull of where is a discrete subgroup of . Thus is an invariant of . Now the double (defined at the start of Subsection 2.1) has the form , where is a Kleinian group containing . Since is hyperbolic with finite volume, is an algebraic number field. Hence the subfield is also an algebraic number field. We compute this by combining Orb with a modified version of Oliver Goodman’s program Snap ([17]).
Snap begins with generators and relations for , and a numerical approximation to provided by Orb. It first refines this using Newton’s method to obtain a high precision numerical approximation to , and then tries to find exact descriptions of matrix entries and their traces as algebraic numbers using the LLLalgorithm. Finally Snap verifies that we have an exact representation of by checking that the relations for are satisfied using exact calculations in a number field, and computes the invariant trace field and associated algebraic invariants. (See [8] for a detailed description of Snap.)
3. Complexity theory
A theory of complexity for 3orbifolds, mimicking Matveev’s theory for manifolds [32], was developed in [41]. Removing all references to edge orders and their contributions to the complexity, one deduces a theory of complexity for 3valent graphs embedded in closed orientable manifolds. In this paragraph we will summarize the main features of this theory. The main ideas of this theory are as follows:

Triangulations are the best way to manipulate 3dimensional topological objects by computer.

Therefore, the minimal number of tetrahedra required to triangulate an object gives a very natural measure of the complexity of the object.

However, there exists another definition of complexity, based on the notion of simple spine. A triangulation, via a certain “duality,” gives rise to a simple spine, therefore complexity defined via spines is not greater than complexity defined via triangulations.

Simple spines are more flexible than triangulations. In particular, there are more general nonminimality criteria for simple spines than for triangulations. More specifically, there are instances where a triangulation may appear to be minimal (as a triangulation) whereas the dual spine is obviously not minimal (as a simple spine).

A theorem ensures that for a hyperbolic object a minimal simple spine is always dual to a triangulation.

As a conclusion, if one wants to carry out a census of hyperbolic objects in order of increasing complexity, one deals by computer with triangulations, but one discards triangulations to which, via duality, the stronger nonminimality criteria for spines apply. This is because, thanks to the theorem, such a triangulation encodes either a nonhyperbolic object or a hyperbolic object that has been met earlier in the census.
We will now turn to a more detailed discussion.
3.1. Simple spines and complexity
To proceed with the key notions and results we recall a definition given in the Introduction. We call simple^{1}^{1}1In [32] such a polyhedron was originally called almost simple, while the term simple was employed for almost special polyhedra, see Subsection 3.2. a compact polyhedron (in the PL sense [44]) such that the link of each point is a subset of the 1skeleton of the tetrahedron. We denote by the set of points of having the whole 1skeleton of the tetrahedron as a link, and we note that is a finite set.
Definition 3.1.
A simple spine of a trivalent graph is a simple polyhedron embedded in in such a way that:

intersects transversely. (In particular, consists of a finite number of points that are not vertices of ).

Removing an open regular neighbourhood of from gives a finite collection of balls, each of which intersects in either

the empty set, or

a single unknotted arc of , or

a vertex of with unknotted strands leaving the vertex and reaching the boundary of the ball. (See Figure 1.)

It is very easy to see (and it will follow from the duality with triangulations in Proposition 3.2) that each admits simple spines. Therefore the complexity of , that we define as
is a finite number.
3.2. Special spines and duality
To illustrate the relation between spines and triangulations, we need to introduce two subsequent refinements of the notion of simple polyhedron. We will say that is almostspecial if it is a compact polyhedron and each of its points has one of the following sets as a link:

The skeleton of the tetrahedron with two open opposite edges removed (a circle);

The skeleton of the tetrahedron with one open edge removed (a circle with a diameter);

The skeleton of the tetrahedron (a circle with three radii).
The corresponding local structure of an almostspecial polyhedron is shown in Figure 6.
Besides the set of vertices already introduced above for simple polyhedra, we can define for an almostspecial the singular set, given by the nonsurface points and denoted by . We remark that is a 4valent graph with vertex set . Note also that if is an almostspecial spine of , by the transversality assumption, intersects away from .
An almostspecial polyhedron is called special if is a union of open discs and is a union of open segments. A special spine of a graph is a simple spine which, in addition, is a special polyhedron.
The following result, which refers to the case of manifolds without graphs embedded in them, has been known for a long time. We point out that we use the term triangulation for a (closed, connected, orientable) manifold in a generalized (not strictly PL [44]) sense. Namely, we mean a realization of as a simplicial pairing between the faces of a finite union of tetrahedra, i.e., we allow multiple and selfadjacencies between tetrahedra.
Proposition 3.2.
Given a manifold , for each triangulation of define as the skeleton of the cell decomposition dual to , see Figure 7.
Then defines a bijection between the set of (isotopy classes of) triangulations of and the set of (isotopy classes of) special spines of .
3.3. (Efficient) triangulations of graphs
We now turn to graphs , and we define a triangulation of to be a (generalized) triangulation of which contains as a subset of its 1skeleton. We will further say that is efficient if it has precisely one vertex at each vertex of , one on each knot component of , and no other vertices.
The following easy result shows that under suitable conditions Proposition 3.2 has a refinement to graphs:
Proposition 3.3.
For a simple spine of a graph the following conditions are equivalent:

is dual to a triangulation of ;

is special, intersects transversely away from , and each component of intersects at most once.
3.4. Minimal spines
A simple spine of a graph is called minimal if it has vertices and no subset of is also a spine of . The success of the strategy based on complexity theory (as outlined at the beginning of this section) for the enumeration of hyperbolic graphs depends on the next three results. They require the concept of irreducibility defined in the introduction. The first one is part of Theorem 2.4, the next two easily follow from [41, Theorem 2.6].
Proposition 3.4.
If is hyperbolic with parabolic meridians then is irreducible.
Proposition 3.5.
The irreducible graphs with are those described as follows and illustrated in Figure 8:

is either , or , or , and is either empty or the core of a Heegaard solid torus of ;

is and is the trivially embedded graph.
Theorem 3.6.
Let be a graph with . Then the following are equivalent:

is irreducible;

admits a special minimal spine;

Every minimal spine of is special and dual to it there is an efficient triangulation of .
3.5. Nonminimality criteria
The following result was used for the enumeration of candidate triangulations of irreducible graphs, as explained in more detail in the next section:
Proposition 3.7.
Let be a triangulation of a graph , and let be the special spine dual to . Suppose that in there is an edge not lying in and incident to distinct tetrahedra, with . Then is not minimal.
Proof.
We will show that we can perform a move on leading to a simple spine of with fewer vertices than .
For we do not even need to use spines, the move exists already at the level of triangulations: it is the famous MatveevPiergallini move [33, 42] illustrated in Figure 9. We only need to note that after the move we still have a triangulation of because the edge that disappears with the move does not lie in .
For we do need to use spines. The moves we apply (a and a move) are illustrated in Figure 10. Both moves involve the removal of the component of dual to the edge of the statement, and the result of the move is still a spine of because does not meet . We note that the move leads to an almostspecial polyhedron, but it can create a spine with an annular nonsingular component, in which case the spine is not dual to a triangulation. The move gives a spine which is not almostspecial. ∎
Remark 3.8.
Sometimes the nonminimality criteria of the previous proposition do not apply directly, but only after a modification of the triangulation. For instance, a triangulation with tetrahedra may be transformed into one with tetrahedra via a move: if contains an edge incident to or distinct tetrahedra, the dual spine can be transformed into a simple spine with at most vertices by applying one of the moves in Figure 10. Therefore the original triangulation is not minimal.
3.6. Complexity of the complement
Matveev’s complexity [32] is defined for every compact manifold, with or without boundary. The complement of an open regular neighbourhood of a graph in a closed manifold therefore has a complexity, which is related to as follows.
Proposition 3.9.
For any graph we have
If is irreducible with and , then
Proof.
If is a minimal simple spine of then the graph intersects in a finite number of points. Removing from open regular neighbourhoods of these points gives a simple polyhedron which is a spine of with the same vertices as . Therefore .
If is irreducible, and , then Theorem 3.6 shows that a minimal simple spine of is special and consists of some points belonging to the interior of distinct disc components of . Removing these discs we get a simple spine of with strictly fewer vertices than . ∎
Remark 3.10.
A compact manifold which admits a complete hyperbolic metric with geodesic boundary and finite volume (after removing the tori from its boundary) has complexity at least , see [32, 2, 11]. This explains why the first hyperbolic knots have (see Tables 1 and 2). Analogously, the first graphs whose complement is hyperbolic with geodesic boundary must have . (In fact they have complexity , see Subsection 5.4.)
4. Computer programs and obstructions to hyperbolicity
In this section we describe the Haskell code we have written to enumerate triangulations, and the computer program Orb we have used to investigate hyperbolic structures. We also describe how nonhyperbolic graphs were identified (see also Section 6 below).
4.1. Enumeration of marked triangulations
Thanks to Theorem 3.6 and the other results stated in the previous section, the enumeration of irreducible graphs of complexity can be performed by listing all efficient triangulations with tetrahedra satisfying some minimality criteria. This was done via a separate program, written in Haskell [18], which suitably adapts the strategy already used in similar censuses (e.g. [30, 35]).
A triangulation of a graph can be encoded as a triangulation of with some marked edges. A triangulation here is just a gluing of tetrahedra, which can be described via a connected valent graph (the incidence graph of the gluing) having a label on each edge encoding how the corresponding triangular faces are identified (there are possibilities).
A first count gives triangulations to check, where is the number of valent graphs with vertices (and edges), shown in Table 3. On each triangulation there are distinct markings of edges, where is the number of edges and is the number of vertices in the triangulation of . Since there are at least triangles in the link of each vertex, , and . There are therefore up to marked triangulations to check. This number is already too big for , so in order to simplify the problem we used some tricks.
We are only interested in orientable manifolds . We can therefore orient each tetrahedron and require the identifications of faces to be orientationreversing. This reduces the number of possible labels on edges from to , and the number of triangulations to , where is the number of valent graphs with “oriented” vertices: each vertex has a fixed parity of orderings of the incident edges. For a fixed valent graph with vertices, the vertices can be oriented in different ways, but up to the symmetries of the number of distinct orientations typically turns out to be very small. This explains why is actually much less than , as shown in the table.
We selected from the resulting list of triangulations only those yielding closed manifolds. Finally, on each triangulation we a priori had distinct markings on edges to analyze. Proposition 3.7 was used to discard many of these: in a triangulation dual to a minimal spine an edge incident to at most distinct tetrahedra is necessarily marked. It remained then to check which markings give rise to efficient triangulations.
4.2. “Orb”
Hyperbolic structures were computed using the program Orb written by Damian Heard [20, 19]. This program builds on ideas of Thurston, Weeks, Casson and others to find hyperbolic structures and associated geometric invariants for a large class of 3dimensional manifolds and orbifolds. The program begins with a triangulation of the space with the singular locus or graph contained in the 1skeleton and tries to find shapes of generalized hyperbolic tetrahedra (with vertices inside, on, or outside the sphere at infinity) which fit together to give a hyperbolic structure.
The generalized hyperbolic tetrahedra are described by using one parameter for each edge in the triangulation. For a general tetrahedron a lift to Minkowski space is chosen, then the parameters are Minkowski inner products of the vertex positions. For compact tetrahedra, each parameter is just the hyperbolic cosine of the edge length. For each ideal vertex the lift to Minkowski space determines a horosphere centred at the vertex; for each hyperideal vertex a geodesic plane orthogonal to the incident faces is determined. Then the edge parameters are simple functions of the hyperbolic distances between these surfaces.
Given the edge parameters, all dihedral angles of the tetrahedra are determined. Moreover the parameters give a global hyperbolic structure if and only if the sum of the dihedral angles around each edge is (or the desired cone angle, in the orbifold case). This gives a system of equations that Orb solves numerically using Newton’s method, starting with suitable regular generalized tetrahedra as the initial guess.
Once a hyperbolic structure is found, Orb can compute many geometric invariants including volumes, the Kojima canonical decompositions, and symmetry groups. This uses methods based on ideas of Weeks [52], Ushijima [50] and FrigerioPetronio [12], too complicated to be reproduced here.
After computing hyperbolic structures numerically using Orb, we checked the correctness of the results by using Jeff Weeks’ program SnapPea [51] to calculate complete hyperbolic structures on the manifolds with torus cusps obtained by doubling along all 3punctured sphere boundary components.
Finally, we verified the results by using Oliver Goodman’s program Snap [17, 8] to find exact hyperbolic structures. This provides a proof that the hyperbolic structures are correct and allows us to compute associated arithmetic invariants (including invariant trace fields), as already mentioned in Subsection 2.6 above.
4.3. Nonhyperbolic knots and links
Many knots and links in the census turned out to be torus links in lens spaces, see Subsection 6.1 below. From , we then decided to rule out the nonhyperbolic knots and links from our census (except for those in at ); this helped a lot in simplifying the classification. Many nonhyperbolic knots and links were easily identified by the following criterion:
Remark 4.1.
The remaining knots and links were shown to be nonhyperbolic by examining their fundamental groups with the help of the following observations.
Lemma 4.2.
Let be an orientable finite volume hyperbolic manifold, and let . Then

if for some integers then ,

if and then .
Proof.
The results are clear if or is the identity, so we may assume that and correspond to loxodromic or parabolic isometries of .
In part (i), the elements must have the same axis or fixed point at . Since the same is true for and , so and commute.
In part (ii), and have the same fixed point set on the sphere at infinity, and takes to itself. Since is not elliptic, it must fix each point of . Thus has the same axis or fixed point at as and , so it commutes with them.∎
Lemma 4.3.
Let be an orientable finite volume hyperbolic manifold. Then cannot have a presentation of the form

where are integers with , or

.
Proof.
(i) If , then by part (i) of Lemma 4.2. Hence and , again by part (i) of Lemma 4.2. So the group would be abelian, which is impossible.
(ii) The group has a presentation
We can rewrite this as
Hence by part (ii) of Lemma 4.2 and . So the group would be abelian, which is again impossible. ∎
Among the knots and links up to complexity 4 for which Orb did not find a hyperbolic structure, all but one of the complements had a fundamental group with presentation of the form , or . These all correspond to nonhyperbolic links by the Lemmas above. The one remaining knot had a presentation as in part (ii) of Lemma 4.3, so is also nonhyperbolic.
4.4. Nonhyperbolic graphs
For graphs with at least one vertex, we first eliminated all triangulations whose dual spines had nonminimal complexity hence were either reducible or occurred earlier in our list. This left a handful of examples for which Orb failed to find a hyperbolic structure. These were first examined using Jeff Weeks’ program SnapPea, by constructing triangulations of the manifolds with torus cusps obtained by doubling along the 3punctured sphere boundary components. We used SnapPea’s “splitting” function to look for incompressible Klein bottles and tori in the doubles. This suggested that incompressible Klein bottles were present in the original graph complements. We then verified this and showed that these examples were indeed nonhyperbolic by theoretical means, as explained below in Section 6.
5. Hyperbolic census details
In this section we will expand on the information given in Table 1, providing details of all the 123 hyperbolic graphs up to complexity 5. Pictures of the hyperbolic graphs up to complexity 4 will be shown in Section 7.
5.1. Name conventions
For future reference, we have chosen a name for each of the graphs we have found. The name has the form
where is the number of vertices of the graph, is a string describing the abstract graph type, is the complexity, and is an index (starting from for any given ). We have found in our hyperbolic census only 6 graph types, described above in Figure 2, so a string of one letter only (or the empty string, for knots) was sufficient to identify them. For graphs with 2 vertices, the letters and were suggested by the common names “graph” and “handcuffs”. The choice of letters was arbitrary for graphs with vertices.
5.2. Organization of tables
We will give separate tables for graphs, handcuffs, vertex graphs, and knots. Within each table, graphs are always arranged in increasing order of their hyperbolic volumes. For graphs having vertices, the columns of the tables respectively contain:

The name of the graph .

The volume of the hyperbolic structure with parabolic meridians on .

A description of the cells of the Kojima canonical decomposition for this structure. When all these cells are tetrahedra we simply indicate their number, otherwise we add an asterisk in the table and provide additional information separately.

The symmetry group of , with denoting the dihedral group with elements.

Whether is chiral (c) or amphichiral (a).

The name of the underlying space . This is almost always a lens space; otherwise it is a Seifert fibred space which we describe in the usual way (as in [34, p. 406]).

The degree, signature and discriminant of the invariant trace field. Details of minimal polynomials for the fields are available on the web at www.ms.unimelb.edu.au/~snap/knotted_graphs.html.

Whether all traces of group elements are algebraic integers.

Whether the group is arithmetic (after doubling to obtain a finite covolume group).
name  volume  (K)  sym  a/c  space  deg  sig  disc  int  ar 

5.333489567  3  c  2  0, 1  7  Y  Y  
5.333489567  3  c  (3,1)  2  0, 1  7  Y  Y  
6.354586557  4  c  3  1, 1  44  Y  N  
6.354586557  4  c  (4,1)  3  1, 1  44  Y  N  
6.551743288  7  c  3  1, 1  107  Y  N  
6.551743288  7  c  (5,2)  3  1, 1  107  Y  N  
6.755194816  5  c  (3,1)  4  0, 2  2917  Y  N  
6.755194816  5  c  (5,1)  4  0, 2  2917  Y  N  
6.927377112  11  c  4  0, 2  1929  Y  N  
6.927377112  11  c  (7,3)  4  0, 2  1929  Y  N  
6.952347978  6  c  (4,1)  5  1, 2  7684  Y  N  
6.952347978  6  c  (6,1)  5  1, 2  7684  Y  N  
6.987763199  7  c  (3,1)  5  1, 2  77041  Y  N  
6.987763199  7  c  (7,2)  5  1, 2  77041  Y  N  
7.035521457  8  c  5  1, 2  5584  Y  N  
7.035521457  8  c  (8,3)  5  1, 2  5584  Y  N  
7.084790037  15  c  5  1, 2  49697  Y  N  
7.084790037  15  c  (9,2)  5  1, 2  49697  Y  N  
7.142157274  9  c  (5,2)  7  1, 3  123782683  Y  N  
7.142157274  9  c  (9,2)  7  1, 3  123782683  Y  N  
7.157517365  8  c  (4,1)  7  1, 3  2369276  Y  N  
7.157517365  8  c  (10,3)  7  1, 3  2369276  Y  N  
7.175425922  9  c  (3,1)  7  1, 3  88148831  Y  N  
7.175425922  9  c  (11,3)  7  1, 3  88148831  Y  N  
7.192635929  11  c  (5,2)  8  0, 4  5442461517  Y  N  
7.192635929  11  c  (11,3)  8  0, 4  5442461517  Y  N  
7.193764490  12  c  7  1, 3  1523968  Y  N  
7.193764490  12  c  (12,5)  7  1, 3  1523968  Y  N  
7.216515907  11  c  (3,1)  8  0, 4  3679703653  Y  N  
7.216515907  11  c  (13,5)  8  0, 4  3679703653  Y  N  
7.327724753  4  a  2  0, 1  4  Y  Y  
7.517689896  6  c  (3,1)  3  1, 1  104  Y  N  
7.706911803  5  c  3  1, 1  59  Y  N  
7.706911803  5  c  (5,1)  3  1, 1  59  Y  N  
7.867901276  7  c  (7,2)  5  3, 1  112919  Y  N  
7.940579248  9  c  (8,3)  3  1, 1  76  Y  N  
7.940579248  9  c  3  1, 1  76  Y  N  
8.000234350  4  c  2  0, 1  7  Y  Y  
8.087973789  5  c  4  2, 1  6724  Y  N  
8.195703083  7  c  (5,2)  5  1, 2  65516  Y  N  
8.233665208  6  c  (6,1)  6  2, 2  1738384  Y  N  
8.338374585  8  c  (9,2)  6  2, 2  2463644  Y  N  
8.355502146  8  c  4  0, 2  3173  Y  N  
8.355502146  6  c  4  0, 2  3173  Y  N  
8.372209945  8  c  (10,3)  7  3, 2  87357184  Y  N  
8.388819035  10  c  (4,1)  5  1, 2  26084  Y  N 
name  volume  (K)  sym  a/c  space  deg  sig  disc  int  ar 

8.403864479  10  c  (11,3)  7  3, 2  186794473  Y  N  
8.487060022  8  c  (9,2)  8  4, 2  17112324248  Y  N  
8.527312899  10  c  (11,3)  9  5, 2  5328053407637  Y  N  
8.546347793  11  c  (12,5)  8  4, 2  2498992192  Y  N  
8.565387019  12  c  (13,5)  9  5, 2  1944699708173  Y  N  
8.612415201  1*  c  (4,1)  4  2, 1  400  Y  N  
8.778658803  9  c  5  1, 2  15856  Y  N  
8.778658803  9  c  5  1, 2  15856  Y  N  
8.793345604  7  c  4  0, 2  257  Y  N  
8.806310033  8  c  (8,3)  4  2, 1  1968  Y  N  
8.908747390  11  c  (3,1)  5  1, 2  31048  Y  N  
8.929317823  6  c  3  1, 1  116  Y  N  
8.967360849  7  c  4  0, 2  697  Y  N  
8.967360849  7  c  (7,2)  4  0, 2  697  Y  N  
9.045557688  5  c  (3,1)  5  1, 2  73532  Y  N  
9.272866192  7  c  6  0, 3  4319731  Y  N  
9.353881135  7  c  (3,1)  6  0, 3  2944468  Y  N  
9.437583617  9  c  4  0, 2  2312  Y  N  
9.491889687  5  c  4  0, 2  257  Y  N  
9.491889687  5  c  (3,1)  4  0, 2  257  Y  N  
9.503403931  9  c  4  0, 2  788  N  N  
10.149416064  1*  c  2  0, 1  3  Y  Y  
10.396867321  6*  c  3  1, 1  139  Y  N  
10.666979134  6  a  2  0, 1  7  Y  Y  
10.666979134  6  c  2  0, 1  7  N  N  
10.666979134  5  c  (3,1)  2  0, 1  7  Y  Y  
10.666979134  5  c  (3,1)  2  0, 1  7  Y  Y 
name  volume  (K)  sym  a/c  space  deg  sig  disc  int  ar 

_ 1_ 1  3.663862377  1  a  2  0, 1  4  Y  Y  
_ 2_ 1  5.074708032  2  a  2  0, 1  3  Y  Y  
_ 3_ 1  5.875918083  3  c  3,1)  4  0, 2  656  Y  N  
_ 3_ 2  6.138138789  5  c  4  0, 2  320  Y  N  
_ 4_ 1  6.354586557  4  c  4,1)  3  1, 1  44  Y  N  
_ 4_ 2  6.559335883  5  c  3,1)  6  0, 3  382208  Y  N  
_ 5_ 1  6.647203159  5  c  5,1)  6  0, 3  242752  Y  N  
_ 4_ 3  6.784755787  9  c  6  0, 3  108544  Y  N  
_ 4_ 4  6.831770496  6  c  4  0, 2  892  Y  N  
_ 5_ 2  6.854770090  7  c  5,2)  8  0, 4  502248448  Y  N  
_ 5_ 3  6.952347978  6  c  4,1)  5  1, 2  7684  Y  N  
_ 5_ 4  6.969842840  5  a  5,2)  6  0, 3  179776  Y  N  
_ 5_ 5  7.008125009  9  c  5,2)  10  0, 5  1192884600832  Y  N  
_ 5_ 6  7.020614792  13  c  8  0, 4  89276416  Y  N  
_ 5_ 7  7.056979121  7  c  3,1)  10  0, 5  586177642496  Y  N  
_ 5_ 8  7.136868364  10  c  6  0, 3  682736  Y  N  
_ 5_ 9  7.146107337  9  c  3,1)  12  0, 6  8746362208256  Y  N  
_ 3_ 3  7.327724753  4  a  2  0, 1  4  Y  Y  
_ 4_ 5  7.327724753  4  a  2  0, 1  4  Y  Y  
_ 5_ 10  7.731874058  5  c  4,1)  6  0, 3  96512  Y  N  
_ 5_ 11  8.140719221  6  c  6  0, 3  382208  Y  N  
_ 5_ 12  8.140719221  5  c  6  0, 3  382208  Y  N  
_ 4_ 6  8.738570409  4  a  4  0, 2  144  Y  N  
_ 5_ 13  8.997351944  3*  c  4  0, 2  784  Y  N  
_ 4_ 7  8.997351944  4  c  4  0, 2  784  Y  N  
_ 4_ 8  8.997351944  4  c  3,1)  4  0, 2  784  Y  N  
_ 5_ 14  9.539780459  5  c  3,1)  4  0, 2  656  Y  N  
_ 5_ 15  9.539780459  5  c  4  0, 2  656  Y  N  
_ 5_ 16  9.592627932  6  c  4  0, 2  1436  Y  N  
_ 5_ 17  9.802001166  5  c  4  0, 2  320  N  N  
_ 5_ 18  9.876829057  5  c  6  0, 3  239168  Y  N  
_ 5_ 19  10.018448934  5  c  6  0, 3  30976  N  N  
_ 5_ 20  10.018448934  5  c  4,1)  6  0, 3  30976  Y  N  
_ 5_ 21  10.018448934  5  c  4,1)  6  0, 3  30976  Y  N  
_ 5_ 22  10.069070958  7  c  4  0, 2  1384  Y  N  
_ 5_ 23  10.149416064  4*  c  2  0, 1  3  Y  Y  
_ 5_ 24  10.215605665  5  c  6  0, 3  732736  N  N  
_ 5_ 25  10.215605665  5  c  5,2)  6  0, 3  732736  Y  N  
_ 5_ 26  10.215605665  5  c  5,2)  6  0, 3  732736  Y  N  
_ 5_ 27  10.408197599  5  c  4  0, 2  441  N  N 
name  volume  (K)  sym  a/c  space  deg  sig  disc  int  ar 
_ 2_ 1  7.327724753  2  a  2  0, 1  4  Y  Y  
_ 5_ 1  11.751836165  6  c  4  0, 2  656  Y  N  
_ 5_ 2  12.661214320  5  c  4  0, 2  784  Y  N  
_ 4_ 1  10.149416064  4  a  2  0, 1  3  Y  Y  
_ 4_ 1  10.991587130  4  a  2  0, 1  4  Y  Y 
5.3. Table of knots
As already mentioned, we have classified hyperbolic knots only up to complexity 4, finding 5 of them. The table containing their description differs from the previous ones only in that the third column gives the number of cells in the EpsteinPenner [9] canonical decomposition (the Kojima decomposition is not defined). We also provide an additional table showing the name of each knot complement in the SnapPea census [2], and either the name of the knot in [43] (for the knots in ) or the surgery coefficients on one of the components of the Whitehead link ( in [43]) yielding the knot.
As shown in the introduction and in Section 6 below, there are many irreducible knots in complexity up to , and most of them are not hyperbolic: this phenomenon can be understood using spines, see Proposition 3.9.
name  volume  (EP)  sym  a/c  space  deg  sig  disc  int  ar 

0_ 3_ 1  2.029883213  2  a  2  0,1  3  Y  Y  
0_ 4_ 1  2.029883213  2  c  2  0,1  3  Y  Y  
0_ 4_ 2  2.568970601  4  c  3  1,1  59  Y  N  
0_ 4_ 3  2.666744783  3  c  2  0,1  7  Y  Y  
0_ 4_ 4  2.828122088  4  c  3  1,1  59  Y  N 
5.4. Compact totally geodesic boundary
The 3 graphs referred to in Proposition 1.1 are , and in Table 5; these are shown in Figure 3. (In particular, Thurston’s knotted [46, pp. 133137] is .) Their hyperbolic structures were constructed using Orb. They all have the lowest possible volume () for hyperbolic manifolds with genus 2 boundary (see [26]), but they can be distinguished by their Kojima decompositions or symmetry groups. All the other graphs were shown not to have such a structure by studying spines for their complements constructed as in the proof of Proposition 3.9. In all but two cases, this produced a spine for the complement of complexity having less than vertices, hence the complement has no hyperbolic structure with geodesic boundary by Remark 3.10. For the two remaining cases, we found a spine having 2 vertices but not dual to a triangulation. It again follows that these manifolds are not hyperbolic with geodesic boundary, because a minimal simple spine of a hyperbolic manifold is always dual to a triangulation [34].
6. Irreducible nonhyperbolic graphs
This section is devoted to the description of the irreducible but nonhyperbolic graphs we have found in our census, including the proof that indeed they have these properties.
6.1. Knots and links
As already stated in the introduction, we have shown that if a graph with is irreducible but nonhyperbolic then has no vertices. More precisely, is either empty, or a knot, or a twocomponent link. Since this paper is chiefly devoted to the understanding of graphs with vertices, we will only very briefly describe our discoveries for the case without vertices. In particular, we will not refer to the case of empty (i.e., to the case of manifolds), addressing the reader to [34], and we will describe the following nonhyperbolic knots and links:

up to complexity , in general manifolds;

in complexity , in .
To proceed we will introduce some general machinery.
6.2. Torus knots in lens spaces
Consider the solid torus and the basis of given by a longitude and a meridian . These elements are characterized up to symmetries of by the property that the restriction to of the map is surjective, while is the kernel of this map.
For coprime